Modelling integer programming with logic: Language and implementation

The classical algebraic modelling approach for integer programming (IP) is not suitable for some real world IP problems, since the algebraic formulati ons allow only for the description of mathematical relations. not logical r elations. In this paper, we present a language L+ for IF, in which we write logical specification of an IP problem. L+ is a language based on the pred icate logic, but is extended with meta predicates such as at least(m, S), w here m is a non-negative integer, meaning that at least m predicates in the set S of formulas hold. The meta predicates facilitate reasoning about a m odel of an IP problem rigorously and logically. L+ is executable in the sen se that formulas in L+ are mechanically translated into a set of mathematic al formulas, called IP formulas, which most of existing IP solvers accept. We give a systematic method for translating formulas in Ci to IP formulas. The translation is rigorously defined, verified and implemented in Mathemat ica 3.0. Our work follows the approach of McKinnon and Williams, and elabor ated the language in that (1) it is rigorously defined, (2) transformation to IP formulas is more optimised and verified, and (3) the transformation i s completely given in Mathematica 3.0 and is integrated into IP solving env ironment as a tool for IF.